Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,4,Mod(1,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(10.3253342510\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 17\nu^{2} + 41\nu + 2 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 25\nu^{2} + 49\nu + 90 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 25\nu^{2} + 11\nu + 74 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 4\beta_{3} + 19\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{4} - 29\beta_{3} + 25\beta_{2} + 33\beta _1 + 209 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.31366 | −1.93939 | 20.2350 | 0 | 10.3052 | −7.00000 | −65.0123 | −23.2388 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.85474 | −8.98858 | 0.149548 | 0 | 25.6601 | −7.00000 | 22.4110 | 53.7945 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.55528 | 4.96149 | −5.58112 | 0 | −7.71648 | −7.00000 | 21.1224 | −2.38365 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.67516 | 2.49396 | −5.19383 | 0 | 4.17779 | −7.00000 | −22.1018 | −20.7802 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.04851 | −6.52749 | 8.39045 | 0 | −26.4266 | −7.00000 | 1.58074 | 15.6081 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 175.4.a.i | 5 | |
| 3.b | odd | 2 | 1 | 1575.4.a.bq | 5 | ||
| 5.b | even | 2 | 1 | 175.4.a.j | 5 | ||
| 5.c | odd | 4 | 2 | 35.4.b.a | ✓ | 10 | |
| 7.b | odd | 2 | 1 | 1225.4.a.be | 5 | ||
| 15.d | odd | 2 | 1 | 1575.4.a.bn | 5 | ||
| 15.e | even | 4 | 2 | 315.4.d.c | 10 | ||
| 20.e | even | 4 | 2 | 560.4.g.f | 10 | ||
| 35.c | odd | 2 | 1 | 1225.4.a.bh | 5 | ||
| 35.f | even | 4 | 2 | 245.4.b.d | 10 | ||
| 35.k | even | 12 | 4 | 245.4.j.f | 20 | ||
| 35.l | odd | 12 | 4 | 245.4.j.e | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.b.a | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 175.4.a.i | 5 | 1.a | even | 1 | 1 | trivial | |
| 175.4.a.j | 5 | 5.b | even | 2 | 1 | ||
| 245.4.b.d | 10 | 35.f | even | 4 | 2 | ||
| 245.4.j.e | 20 | 35.l | odd | 12 | 4 | ||
| 245.4.j.f | 20 | 35.k | even | 12 | 4 | ||
| 315.4.d.c | 10 | 15.e | even | 4 | 2 | ||
| 560.4.g.f | 10 | 20.e | even | 4 | 2 | ||
| 1225.4.a.be | 5 | 7.b | odd | 2 | 1 | ||
| 1225.4.a.bh | 5 | 35.c | odd | 2 | 1 | ||
| 1575.4.a.bn | 5 | 15.d | odd | 2 | 1 | ||
| 1575.4.a.bq | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 4T_{2}^{4} - 21T_{2}^{3} - 70T_{2}^{2} + 54T_{2} + 160 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(175))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 4 T^{4} + \cdots + 160 \)
|
| $3$ |
\( T^{5} + 10 T^{4} + \cdots + 1408 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( T^{5} - 42 T^{4} + \cdots + 11139072 \)
|
| $13$ |
\( T^{5} + 34 T^{4} + \cdots + 51332920 \)
|
| $17$ |
\( T^{5} + 238 T^{4} + \cdots - 359983792 \)
|
| $19$ |
\( T^{5} + 36 T^{4} + \cdots - 20133120 \)
|
| $23$ |
\( T^{5} + \cdots + 8730849408 \)
|
| $29$ |
\( T^{5} + \cdots + 126081243400 \)
|
| $31$ |
\( T^{5} + \cdots - 34433580800 \)
|
| $37$ |
\( T^{5} + \cdots - 43388412544 \)
|
| $41$ |
\( T^{5} + \cdots + 109849343072 \)
|
| $43$ |
\( T^{5} + \cdots + 49627325440 \)
|
| $47$ |
\( T^{5} + \cdots - 32437777344 \)
|
| $53$ |
\( T^{5} + 976 T^{4} + \cdots + 744413184 \)
|
| $59$ |
\( T^{5} + \cdots + 5009454255200 \)
|
| $61$ |
\( T^{5} + \cdots - 650238065792 \)
|
| $67$ |
\( T^{5} + \cdots - 15885557727232 \)
|
| $71$ |
\( T^{5} + \cdots - 7767441797120 \)
|
| $73$ |
\( T^{5} + \cdots + 9409174483008 \)
|
| $79$ |
\( T^{5} + \cdots + 201540544400 \)
|
| $83$ |
\( T^{5} + \cdots + 2803375323136 \)
|
| $89$ |
\( T^{5} + \cdots + 68855772276480 \)
|
| $97$ |
\( T^{5} + \cdots - 12547593502192 \)
|
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